Most of the cryptographic protocols used in everyday life are
based on number theoretic problems such as integer factoring. We will
give an introduction to lattice-based cryptography, a form of
cryptography offering many advantages over the traditional
number-theoretic-based ones, including conjectured security against
quantum computers. The talk will mainly focus on the so-called Learning
with Errors (LWE) problem. This problem has turned out to be an
amazingly versatile basis for cryptographic constructions, with hundreds
of applications, including recent breakthrough work on fully homomorphic
encryption by Gentry and others. In addition to applications, we will
also mention work on using algebraic number theory for making
cryptographic constructions more efficient, as well as some very recent
work on quantum algorithms for related algebraic problems.